6/26/2014
COLLEGE PHYSICS
Chapter 3: Two-Dimensional
Kinematics
Lesson 7
Video Narrated by Jason Harlow,
Physics Department, University of Toronto
VECTORS
 A quantity having both a
magnitude and a direction is
called a vector
 The geometric
representation of a vector is
an arrow with the tail of the
arrow placed at the point
where the measurement is
made
1
6/26/2014
BUT FIRST … REVIEW!
RIGHT TRIANGLE TRIGONOMETRY
This is one of the most common things people are
rusty with…
 sin 𝜃 =
 cos 𝜃 =
 tan 𝜃 =
opp
hyp
SOH CAH TOA
Opposite
adj

hyp
Adjacent
opp
adj
GIVE IT A TRY!
If the hypotenuse shown is 6m and the angle is
38, what is the length of the adjacent side?
A.
6 m sin(38°)
B.
6 m tan(38°)

C.
6 m cos(52°)
Adjacent
D.
6 m cos(38°)
Opposite
E. Not enough information
2
6/26/2014
GIVE IT A TRY!
If the adjacent side below is 6 m and the opposite
side is 4 m, what is the angle  ?
A. tan−1 (6/4)
B. cos −1 (6/4)
C. cos
−1
Opposite
(4/6)

D. tan (4/6)
Adjacent
−1
E. sin−1 (4/6)
Walkers and drivers in a city like New York are rarely
able to travel in straight lines to reach their
destinations.
Instead, they must follow roads and sidewalks, making
two-dimensional, zigzagged paths.
3
6/26/2014
PROPERTIES OF VECTORS
 Suppose a pedestrian walks 9 blocks east, then turns
left and walks 5 blocks north.
 In this city, every block is a 100 m x 100 m square
𝜃 = tan−1
𝑏
𝑎
𝜃
• The Pythagorean theorem relates the length of the
legs of a right triangle, labeled a and b , with the
hypotenuse, labeled c .
• The relationship is given by: 𝑎2 + 𝑏 2 = 𝑐 2 .
• The angle of c is given by: tan 𝜃 =
𝑏
𝑎
4
6/26/2014
DISPLACEMENT
The displacement is 10.3 blocks (or 1030 m) at
an angle 29.1º north of east.
PROPERTIES OF VECTORS
 Sam and Bill are neighbors
 They both walk 200 m to the
northeast of their own front
doors
 Two vectors are equal if they have the
same magnitude and direction
 This is true regardless of the starting
points of the vectors
B=S
5
6/26/2014
VECTOR ADDITION
GEOMETRIC METHOD 1: TIP-TO-TAIL
 To evaluate F = D + E, you could move E over and
use the tip-to-tail rule
EXPLORE!
VECTOR ADDITION
http://phet.colorado.edu/en/simulation/vector-addition
6
6/26/2014
VECTOR ADDITION
GEOMETRIC METHOD 2: PARALLELOGRAM METHOD
 Alternatively, F = D + E can be found as the diagonal
of the parallelogram defined by D and E
ADDITION OF MORE THAN TWO VECTORS
 The figure shows the path of a hiker moving from initial
position 0 to position 1, then 2, 3, and finally arriving at
position 4
 The hiker’s net displacement, an arrow from position 0 to
4, is:
7
6/26/2014
GIVE IT A TRY!
Which figure shows the vector sum A1 + A2 + A3?
MORE VECTOR MATHEMATICS
• The negative of a
vector is just another
vector of the same
magnitude, but
pointing in the
opposite direction.
• B + (−B) = 0, the
zero-vector
8
6/26/2014
MULTIPLYING A VECTOR BY A SCALAR
VECTOR SUBTRACTION
9
6/26/2014
VECTOR SUBTRACTION
GIVE IT A TRY!
Which figure shows the vector difference 2A – B?
10
6/26/2014
COMPONENT VECTORS
 The figure shows a
vector A and an xycoordinate system that
we’ve chosen
 We can define two new
vectors parallel to the
axes that we call the
component vectors of A,
such that:
 We have broken A into two perpendicular vectors that
are parallel to the coordinate axes
 This is called the decomposition of A into its
component vectors
COMPONENTS
 We can describe each component vector with a single
number called the component
 The component tells us how big the component vector is,
and, with its sign, which ends of the axis the component
vector points toward
11
6/26/2014
Switching
between
(magnitude,
direction)
notation and
(x-component,
y-component)
notation:
12
6/26/2014
VECTOR ADDITION AND SUBTRACTION: ANALYTICAL
METHOD
 We can perform vector addition by adding the x- and ycomponents separately
 For example, if D = A + B + C, then:
 Similarly, to find R = P – Q we would compute:
 To find T = cS, where c is a scalar, we would compute:
GIVE IT A TRY!
The sum 𝐹1 + 𝐹2 + 𝐹3
points to the left.
Two of three forces are
shown. Which is the
missing third force?
A.
B.
C.
13
6/26/2014
TILTED AXES AND
ARBITRARY
DIRECTIONS
 For some problems it is
convenient to tilt the axes
of the coordinate system
 The axes are still
perpendicular to each
other, but there is no
requirement that the x-axis
has to be horizontal
𝑦
𝑥
𝑨
 Tilted axes are useful if you need to determine
component vectors “parallel to” and “perpendicular to” an
arbitrary line or surface
14